Note on the evaluation of the memory-kernel occuring in generalized master equations

In this communication we comment on a recent work [12] on the evaluation of the memory-kernel of the generalized master equation. We derive in a transparent and straightforward way the basic expression for the memory kernel. We demonstrate that the evaluation of this expression in [12] is carried out by use of the exact Laplace transform of the Greens function solution of the master equation.     

一种基于波动方程的新高阶FDTD方法

 提出了一种基于二阶波动方程的（2M,4）高阶时域有限差分(FDTD)方法，通过使用辛积分传播子(SIP)在时域上获得4阶精度，使用离散奇异卷积(DSC)方法在空域上达到2M阶精度。与已有的(2M，4) 阶FDTD方法相比，虽然两者都采用SIP和DSC方法，但是此二者的不同点在于：第一，新方法基于二阶波动方程；第二，在离散计算空间时使用单一网格而不是传统的Yee网格；第三，单独计算某一场分量从而节约内存并减少计算量。数值计算结果表明，与传统高阶算法相比，基于波动方程的高阶FDTD方法耗费的机时只有它的50%,内存消耗下降10%, 而两者的计算结果之间相对误差小于5‰。     

Memory-efficient iterative process on a two-dimensional first-order regular graph

We present a parallel and memory-efficient iterative algorithm based on 2D first-order regular graphs. For an M x N regular graph with L iterations, a carefully chosen computation order can reduce the memory resources from O(MN) to O(ML). This scheme can achieve a memory reduction of 4 to 27 times in typical computation-intensive problems such as stereo and motion.     

CPU vs. GPU - Performance comparison for the Gram-Schmidt algorithm

The Gram-Schmidt method is a classical method for determining QR decompositions, which is commonly used in many applications in computational physics, such as orthogonalization of quantum mechanical operators or Lyapunov stability analysis. In this paper, we discuss how well the Gram-Schmidt method performs on different hardware architectures, including both state-of-the-art GPUs and CPUs. We explain, in detail, how a smart interplay between hardware and software can be used to speed up those rather compute intensive applications as well as the benefits and disadvantages of several approaches. In addition, we compare some highly optimized standard routines of the BLAS libraries against our own optimized routines on both processor types. Particular attention was paid to the strong hierarchical memory of modern GPUs and CPUs, which requires cache-aware blocking techniques for optimal performance. Our investigations show that the performance strongly depends on the employed algorithm, compiler and a little less on the employed hardware. Remarkably, the performance of the NVIDIA CUDA BLAS routines improved significantly from CUDA 3.2 to CUDA 4.0. Still, BLAS routines tend to be slightly slower than manually optimized code on GPUs, while we were not able to outperform the BLAS routines on CPUs. Comparing optimized implementations on different hardware architectures, we find that a NVIDIA GeForce GTX580 GPU is about 50% faster than a corresponding Intel X5650 Westmere hexacore CPU. The self-written codes are included as supplementary material.     

Wide-angle split-step spectral method for 2D or 3D beam propagation

We develop a method for non-paraxial beam propagation that obtains a speed improvement over the Finite-Difference Split-Step method (FDSSNP) recently reported by Sharma et al. The method works in the eigen-basis of the Laplace operator ${\left(\nabla_T^2\right)}$ , and in general requires half as many operations to propagate one step forward so that a 2X speedup can be realized. However, the new formulation allows the Fast Fourier Transform (FFT) algorithm to be used, which allows an even greater speedup. The method does not require a numerical matrix inversion, diagonalization, or series evaluation. The diffraction operator is not approximated, and in the absence of refractive index fluctuations the method reduces to an exact solution of the Helmholtz equation.     

三维Laplace方程及Poisson方程的有限分析解表达式和它的应用

本文给出三维Laplace方程及Poisson方程有限分析解的正确系数值,并与陈景仁^[1]给出的系数值进行了比较。给出了三维Laplace方程的有限分解表达式。利用本文的系数值求解热传导问題的算例,与用边界单元法^[6]结果相比,基本一致,但程序编制和花费机时却节省很多。     

The reduction of the Laplace equation in certain Riemannian spaces and the resulting Type II hidden symmetries

We prove a general theorem which allows the determination of Lie symmetries of the Laplace equation in a general Riemannian space using the conformal group of the space. Algebraic computing is not necessary. We apply the theorem in the study of the reduction of the Laplace equation in certain classes of Riemannian spaces which admit a gradient Killing vector, a gradient Homothetic vector and a special Conformal Killing vector. In each reduction we identify the source of Type II hidden symmetries. We find that in general the Type II hidden symmetries of the Laplace equation are directly related to the transition of the CKVs from the space where the original equation is defined to the space where the reduced equation resides. In particular we consider the reduction of the Laplace equation (i.e., the wave equation) in the Minkowski space and obtain the results of all previous studies in a straightforward manner. We consider the reduction of Laplace equation in spaces which admit Lie point symmetries generated from a non-gradient HV and a proper CKV and we show that the reduction with these vectors does not produce Type II hidden symmetries. We apply the results to general relativity and consider the reduction of Laplace equation in locally rotational symmetric space times (LRS) and in algebraically special vacuum solutions of Einstein’s equations which admit a homothetic algebra acting simply transitively. In each case we determine the Type II hidden symmetries.     

Analytical expressions for the correlation function of a hard sphere dimer fluid

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.     

Analysis of Millimeter Wave Scattering by the Infinite Plane Metallic Grating Using PCG-FFT Technique

In this paper, both fast Fourier transformation (FFT) and preconditioned CG technique are introduced into method of lines (MOL) to further enhance the computational efficiency of this semi-analytic method. Electromagnetic wave scattering by an infinite plane metallic grating is used as the examples to describe its implementation. For arbitrary incident wave, Helmholz equation and boundary condition are first transformed into new ones so that the impedance matrix elements are calculated by FFT technique. As a result, this Topelitz impedance matrix only requires O(N) memory storage for the conjugate gradient FFT method to solve the current distribution with the computational complexity O(N log N) . Our numerical results show that circulate matrix preconditioner can speed up CG-FFT method to converge in much smaller CPU time than the banded matrix preconditioner.     

解FOKKER-PLANCK-LANDAU方程的谱方法

利用谱方法和FFT技术对等离子体中带电粒子输运Fokker-Planck-Landau方程进行数值求解,研究空间均匀条件下粒子相空间分布函数随时间的演化.数值计算表明,所用计算格式能够很好地满足粒子数、动量和能量守恒要求,FFT技术的采用也使得运算工作量大为降低.     

三维粗糙面电磁双站散射的直接型区域分解计算

提出三维粗糙面双站电磁散射的直接型有限元-区域分解方法.首先建立含有迭代Robin边界条件(IRBC)的区域分解法耦合模型,再用内视法导出高度稀疏分块的分区耦合矩阵,之后给出缩减耦合矩阵带宽的子区域排序方法和IRBC的FFT加速算法.用有限元-完全匹配层和未分区的有限元-IRBC方法验证数值结果.     

核磁自旋回波串的液体分量分解快速反演法(英文)

该文叙述核磁自旋回波串的液体分量分解快速反演法.此方法假定液体,无论是在散装形式或饱和多孔介质中,可以用一个或一组核磁弛豫线形来表征.对一维核磁共振的拉普拉斯反演,它可以是预先确定的一个或一组T2或T1分布.对二维核磁共振的拉普拉斯反演,它可以是一个或一组预先确定的( D, T2)或( T1, T2)二维分布.对三维核磁共振的拉普拉斯反演,它可以是一个或一组预先设定的( D, T1, T2)三维分布.这些预先确定的线形,可以是高斯、B样条或预先由实验或经验确定的任何线形.这种方法可以显着降低核磁共振数据反演的计算时间,特别是从石油核磁共振测井采集的多维数据反演,它不需牺牲反演所得的分布的平滑性和准确性.这种方法的另一个新应用是作为一种约束求解方法来过滤相邻深度所采集的数据噪音.核磁共振测井的噪音信号,往往造成在相邻深度的同一岩性岩层有不同的T2分布.在此情况下, T2分布就不能用来识别岩性.通过非一般的矩阵操作,作者成功实现了对相邻深度的回波串实施约束求解方法,从而使得T2分布成为一种可靠的岩性识别指标.     

Exact solutions of the Klein—Gordon equation with ring-shaped oscillator potential by using the Laplace integral transform

We present exact solutions for the Klein-Gordon equation with a ring-shaped oscillator potential. The energy eigenvalues and the normalized wave functions are obtained for a particle in the presence of non-central oscillator potential. The angular functions are expressed in terms of the hypergeometric functions. The radial eigenfunctions have been obtained by using the Laplace integral transform. By means of the Laplace transform method, which is efficient and simple, the radial Klein-Gordon equation is reduced to a first-order differential equation.     

Matrix continued fraction solutions of the Kramers equation and their inverse friction expansions

The distribution function in position and velocity space for the Brownian motion of particles in an external field is determined by the Kramers equation, i.e., by a two variable Fokker-Planck equation. By expanding the distribution function in Hermite functions (velocity part) and in another complete set satisfying boundary conditions (position part) the Laplace transform of the initial value problem is obtained in terms of matrix continued fractions. An inverse friction expansion of the matrix continued fractions is used to show that the first Hermite expansion coefficient may be determined by a generalized Smoluchowski equation. The first terms of the inverse friction expansion of this generalized Smoluchowski operator and of the memory kernel are given explicitly. The inverse friction expansion of the equation determining the eigenvalues and eigenfunctions is also given and the connection with the result of Titulaer is discussed.     

A quantum search algorithm of two entangled registers to realize quantum discrete Fouriertransform of signal processing

The discrete Fourier transform （DFT） is the base of modern signal processing. 1-dimensional fast Fourier transform （1D FFT） and 2D FFT have time complexity O（N log N） and O（N^2 log N） respectively. Since 1965, there has been no more essential breakthrough for the design of fast DFT algorithm. DFT has two properties. One property is that DFT is energy conservation transform. The other property is that many DFT coefficients are close to zero. The basic idea of this paper is that the generalized Grover＇s iteration can perform the computation of DFT which acts on the entangled states to search the big DFT coefficients until these big coefficients contain nearly all energy. One-dimensional quantum DFT （1D QDFT） and two-dimensional quantum DFT （2D QDFT） are presented in this paper. The quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, 1D and 2D QDFT have time complexity O（v/N） and O（N） respectively. QDFT and quantum convolution demonstrate that quantum computation to process classical signal is possible.     

三维时变等离子体目标的电磁散射特性研究

基于电流密度拉普拉斯变换方法改进的时域有限差分(LTJEC-FDTD)算法, 研究时变等离子体目标的电磁散射特性.由Maxwell方程和等离子体本构方程出发, 利用拉普拉斯变换和拉普拉斯逆变换, 推导出计算三维时变问题的时域有限差分(FDTD)算法的迭代式. 采用模式匹配方法验证了FDTD迭代式的正确性, 并通过计算等离子体球的雷达散射截面(RCS)验证了算法相关边界的正确性. 最后用LTJEC-FDTD算法分析了涂覆时变等离子体的战斧式巡航导弹的RCS. 关键词： 时变等离子体 雷达散射截面 模式匹配方法 时域有限差分方法     

On NUFFT-based gridding for non-Cartesian MRI

For MRI with non-Cartesian sampling, the conventional approach to reconstructing images is to use the gridding method with a Kaiser-Bessel (KB) interpolation kernel. Recently, Sha et al. [L. Sha, H. Guo, A.W. Song, An improved gridding method for spiral MRI using nonuniform fast Fourier transform, J. Magn. Reson. 162(2) (2003) 250-258] proposed an alternative method based on a nonuniform FFT (NUFFT) with least-squares (LS) design of the interpolation coefficients. They described this LS_NUFFT method as shift variant and reported that it yielded smaller reconstruction approximation errors than the conventional shift-invariant KB approach. This paper analyzes the LS_NUFFT approach in detail. We show that when one accounts for a certain linear phase factor, the core of the LS_NUFFT interpolator is in fact real and shift invariant. Furthermore, we find that the KB approach yields smaller errors than the original LS_NUFFT approach. We show that optimizing certain scaling factors can lead to a somewhat improved LS_NUFFT approach, but the high computation cost seems to outweigh the modest reduction in reconstruction error. We conclude that the standard KB approach, with appropriate parameters as described in the literature, remains the practical method of choice for gridding reconstruction in MRI.     

Development of a new approach to simulate a particle track under electrochemical etching in polymeric detectors

A numerical approach based on image processing was developed to simulate a particle track in a typical polymeric detector, e.g., polycarbonate, under electrochemical etching. The physical parameters such as applied voltage, detector thickness, track length, the radii of curvature at the tip of track, and the incidence angle of the particle were considered, and then the boundary condition of the problem was defined. A numerical method was developed to solve Laplace equation, and then the distribution of the applied voltage was obtained through the polymer volume. Subsequently, the electric field strengths in the detector elements were computed. In each step of the computation, an image processing technique was applied to convert the computed values to grayscale images. The results showed that a numerical solution to Laplace equation is dedicatedly an attractive approach to provide us the accurate values of electric field strength through the polymeric detector volume as well as the track area. According to the results, for a particular condition of the detector thickness equal to 445 μm, track length of 21 μm, the radii of 2.5 μm at track tip, the incidence angle of 90°, and the applied voltage of 2080 V, after computing Laplace equation for an extremely high population of 4000 × 4000 elements of detector, the average field strength at the tip of track was computed equal to 0.31 MV cm⁻¹ which is in the range of dielectric strength for polymers. The results by our computation confirm Smythe’s model for estimating the ECE-tracks.     

A FAST TIME-DOMAIN INTEGRATION METHOD FOR COMPUTING NON-STATIONARY RESPONSE HISTORIES OF LINEAR OSCILLATORS WITH DISCRETE-TIME RANDOM FORCING

A new approach is presented for computing displacement histories of single linear oscillators with arbitrarily light damping and general forcing—of particular use for efficient Monte Carlo simulation of modal systems with ultra-light damping and very broadband non-Gaussian excitation. Solution methods are initially presented within a state transition context, to show limitations of FFT solutions, and to establish, for long-run non-stationary stochastic analysis via fast Laplace, the need for appropriate zero-padding, high cut-off frequency, and fixed-step sampling. Truncation errors arising in single-transition time-domain convolution are then examined via the Euler-Maclaurin summation formula. Errors are shown to be minimum when transition intervals are chosen as integer multiples of the damped natural period, precisely where the O (Δτ²Δf′) error can be evaluated, and the velocity transition equation can be dispensed with. The paper shows that an optimum O (Δτ⁴Δf?) integration scheme can be used for fast time-domain convolution in a two-stage algorithm. First, phased-pairs of accurate displacements are efficiently predicted at selected transition times. These are then used as boundary conditions in adaptive Chebychev polynomial solution giving continuous displacement histories for selected cycles—this considerably reduces the number of multiplications and integrations normally required. Two-stage integration turns out to be at least 100 times faster than explicit short-transition time-domain solution, and for general applications, at least as fast as the Laplace/IFFT approach. But for non-stationary probability density estimation, involving far-future history prediction the speed advantage over fast Laplace can be enormous.     

Memory effects in the diffusion of an interacting polydisperse suspension: I. Projection formalism at long wavelength

We consider collective diffusion and self-diffusion in a polydisperse suspension of macroparticles. Using the generalized Smoluchowski equation to describe the interacting suspension we extend Ackerson's work to derive a multispecies projection formalism for the density fluctuations. We include 2-body direct potential interactions as well as accurate 2-body hydrodynamic interactions. We take the long wavelength low density limit of this formalism to obtain an expression for the memory matrix expressed in terms of 2-body interactions and propagators. We show how memory contributions can arise to first order in density from the fact that the correct mobility tensors are not divergenceless.